Designs for computer experiments and uncertainty quantification
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Computer experiments are widely-used in analysis of real systems where physical experiments are infeasible or unaffordable. Computer models are usually complex and computationally demanding, consequently, time consuming to run. Therefore, surrogate models, also known as emulators, are fitted to approximate these computationally intensive computer models. Since emulators are easy-to-evaluate they may replace computer models in the actual analysis of the systems. Experimental design for computer simulations and modeling of simulated outputs are two important aspects of building accurate emulators. This thesis consists of three chapters, covering topics in design of computer experiments and uncertainty quantification of complex computer models. The first chapter proposes a new type of space-filling designs for computer experiments, and the second chapter develops an emulator-based approach for uncertainty quantification of machining processes using their computer simulations. Finally, third chapter extends the experimental designs proposed in the first chapter and enables to generate designs with both quantitative and qualitative factors. In design of computer experiments, space-filling properties are important. The traditional maximin and minimax distance designs consider only space-fillingness in the full-dimensional space which can result in poor projections onto lower-dimensional spaces, which is undesirable when only a few factors are active. On the other hand, restricting maximin distance design to the class of Latin hypercubes can improve one-dimensional projections but cannot guarantee good space-filling properties in larger subspaces. In the first chapter, we propose designs that maximize space-filling properties on projections to all subsets of factors. Proposed designs are called maximum projection designs. Maximum projection designs have better space-filling properties in their projections compared to other widely-used space-filling designs. They also provide certain advantages in Gaussian process modeling. More importantly, the design criterion can be computed at a cost no more than that of a design criterion which ignores projection properties. In the second chapter, we develop an uncertainty quantification methodology for machining processes with uncertain input factors. Understanding the uncertainty in a machining process using the simulation outputs is important for careful decision making. However, Monte Carlo-based methods cannot be used for evaluating the uncertainty when the simulations are computationally expensive. An alternative approach is to build an easy-to-evaluate emulator to approximate the computer model and run the Monte Carlo simulations on the emulator. Although this approach is very promising, it becomes inefficient when the computer model is highly nonlinear and the region of interest is large. Most machining simulations are of this kind because the output is affected by a large number of parameters including the workpiece material properties, cutting tool parameters, and process parameters. Building an accurate emulator that works for different kinds of materials, tool designs, tool paths, etc. is not an easy task. We propose a new approach, called in-situ emulator, to overcome this problem. The idea is to build an emulator in a local region defined by the user-specified input uncertainty distribution. We use maximum projection designs and Gaussian process modeling techniques for constructing the in-situ emulator. On two solid end milling processes, we show that the in-situ emulator methodology is efficient and accurate in uncertainty quantification and has apparent advantages over other conventional tools. Computer experiments with quantitative and qualitative factors are prevalent. In the third chapter, we extend maximum projection designs so that they can accommodate qualitative factors as well. Proposed designs maintain an economic run size and they are flexible in run size, number of quantitative and qualitative factors and factor levels. Their construction is not restricted to a special design class and does not impose any design configuration. A general construction algorithm, which utilizes orthogonal arrays, is developed. We have shown on several simulations that maximum projection designs with both quantitative and qualitative factors have attractive space-filling properties for all of their projections. Their advantages are also illustrated on optimization of a solid end milling process simulation. Finally, we propose a methodology for sequential construction of maximum projection designs which ensures efficient analysis of systems within financial cost and time constraints. The performance of the sequential construction methodology is demonstrated using the optimization of a solid end milling process.